The former Cardiff and Middlesbrough player has found chances to impress under Roy Keane at the Stadium of Light difficult since moving from Wigan last season.
Kavanagh, a former Republic of Ireland international, is famed for his scoring rate from midfield and tireless work ethic.
He was a regular for Wigan as The Latics gained promotion to the Premiership and did well in their first season in the top flight before moving back to the north east last season.
It is rumoured that The Owls want to sign Kavanagh on loan, whether it is a short-term move or a longer deal with a view to a permanent switch is still up in the air.
Leeds United have been chasing the 34-year-old although reports indicate that the grey-haired star would favour a move to the Championship.
Meanwhile, Sunderland will honour club legend Ian Porterfield, a former Owls player in the seventies, at their home game against Reading on Saturday.
Porterfield, who scored the winner in the 1973 FA Cup Final for Sunderland, died earlier this week aged 61 - Ian was the manager of the Armenian national team at the time of his death.